Gaussian Quantum Discord

We extend the quantum discord to continuous variable systems and evaluate Gaussian quantum discord $C(\varrho )$ for bipartite Gaussian states. In particular, for squeezed-thermal states, we explicitly maximize the extractable information over Gaussian measurements: $C(\varrho )$ is minimized by a generalized measurement rather than a projective one. Almost all squeezed-thermal states have nonzero Gaussian discord: They may be either separable or entangled if the discord is below the threshold $C(\varrho )=1$, whereas they are all entangled above the threshold. We elucidate the general role of state parameters in determining the discord and discuss its evolution in noisy channels.

Figure 1

Left: Gaussian discord $C(\varrho )$ for STS at separability threshold as a function of the ratio $N_1/N_T$ for increasing values of the energy $N_T$ (bottom to top $N_T=1$, 5, 10, 50, ${10}^2$, ${10}^3$, and ${10}^5$). Right: Symplectic eigenvalues of the partial transpose ${\tilde{d}}_{-}$ versus $C(\varrho )$ for randomly generated STS. The red region corresponds to symmetric states. The separability threshold is ${\tilde{d}}_{-}=\frac{1}{2}$ and it corresponds to $C(\varrho )=1$ [$C(\varrho )=2\log 2-1$ symmetric STS].

Figure 2

Gaussian discord $C(\varrho )$ for symmetric STS as a function of the global purity $\mu$ of the state $\rho$ and ${\tilde{d}}_{-}$. Left: Separable states; from bottom to top ${\tilde{d}}_{-}=0.5$,0.6, 0.7, 0.8, and 0.9. Right: Entangled states; from bottom to top ${\tilde{d}}_{-}=0.5$, 0.4, 0.3, 0.3, 0.1, and 0.062 84. See text for description.